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2xdx-1-x-4-




Question Number 124398 by bounhome last updated on 03/Dec/20
∫((2xdx)/( (√(1+x^4 ))))=?
$$\int\frac{\mathrm{2}{xdx}}{\:\sqrt{\mathrm{1}+{x}^{\mathrm{4}} }}=? \\ $$
Answered by Dwaipayan Shikari last updated on 03/Dec/20
∫((2xdx)/( (√(1+x^4 ))))       x^2 =t⇒2x=(dt/dx)  =∫(dt/( (√(1+t^2 ))))=log(t+(√(t^2 +1)))=log(x^2 +(√(x^4 +1)))+C
$$\int\frac{\mathrm{2}{xdx}}{\:\sqrt{\mathrm{1}+{x}^{\mathrm{4}} }}\:\:\:\:\:\:\:{x}^{\mathrm{2}} ={t}\Rightarrow\mathrm{2}{x}=\frac{{dt}}{{dx}} \\ $$$$=\int\frac{{dt}}{\:\sqrt{\mathrm{1}+{t}^{\mathrm{2}} }}={log}\left({t}+\sqrt{{t}^{\mathrm{2}} +\mathrm{1}}\right)={log}\left({x}^{\mathrm{2}} +\sqrt{{x}^{\mathrm{4}} +\mathrm{1}}\right)+{C} \\ $$

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